Downing Street has hit out at claims that some survivors of the Grenfell Tower fire face being rehoused hundreds of miles away from London.

Number Ten said Prime Minister Theresa May received assurances that the commitment to rehome people within the same or neighbouring borough would be met within a three-week deadline as she chaired a meeting of the Government taskforce overseeing the response to the disaster.

Downing Street said reports of people being offered homes hundreds of miles away were “false”, and that the first rehousing offers had now been made.

The taskforce meeting heard that by the end of Monday the UK’s Department for Communities and Local Government (DCLG) expected to receive figures from all local authorities and housing associations in England on the total number of high-rise buildings across the country which would be subject to additional safety checks after the London disaster.

Downing Street said the DCLG has written to the heads of local authorities providing more details to help councils identify a particular type of cladding which is being subjected to additional checks, and that facilities have been set up for testing of samples from Tuesday.

Mrs May expects to receive “significant updates” on the help given to people affected by the blaze when she hosts a meeting with Cabinet ministers on Wednesday, No 10 said.

The Lord Chief Justice is set to appoint a judge to oversee an independent Inquiry “in the next couple of days”, Downing Street said.

“While it was clear some progress is being made, the Prime Minister will continue to receive daily updates to ensure that the steps taken are being carried forward at sufficient scale and speed that help is getting to people who need it,” a Downing Street spokeswoman said.

The taskforce meeting heard that the Department for Work and Pensions has begun administering the first £5,000 payments from a discretionary fund directly into the bank accounts of eligible households affected by the fire.

Mrs May, who was heavily criticised by some for her initial response to the disaster, has said that support for families who needed help or basic information in the initial hours after the Grenfell Tower fire “was not good enough”.